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Mathematics of Computation

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$ L^{p}$ approximation of Fourier transforms and certain interpolating splines

Author: David C. Shreve
Journal: Math. Comp. 28 (1974), 779-787
MSC: Primary 65T05; Secondary 42A68
MathSciNet review: 0383803
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Abstract: We extend to $ {L^p},1 \leqq p \leqq \infty $, the $ {L^2}$ results of Bramble and Hilbert on convergence of discrete Fourier transforms and on approximation using smooth splines. The main tools are the estimates of [1] for linear functionals on Sobolev spaces and elementary results on Fourier multipliers.

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  • [1] J. H. Bramble & S. R. Hilbert, "Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation," SIAM J. Numer. Anal., v. 7, 1970, pp. 112-124. MR 41 #7819. MR 0263214 (41:7819)
  • [2] J. H. Bramble & S. R. Hilbert, "Bounds for a class of linear functionals with applications to Hermite interpolation," Numer. Math., v. 16, 1970/71, pp. 362-369. MR 44 #7704. MR 0290524 (44:7704)
  • [3] R. E. Edwards, Fourier Series: A Modern Introduction. Vol. II, Holt, Rinehart and Winston, New York, 1967. MR 36 #5588. MR 0222538 (36:5588)
  • [4] L. Hörmander, "Estimates for translation invariant operators in $ {L^p}$ spaces," Acta Math., v. 104, 1960, pp. 93-140. MR 22 #12389. MR 0121655 (22:12389)
  • [5] S. D. Silliman, "The numerical evaluation by splines of the Fourier transform," J. Approximation Theory (To appear.) MR 0356556 (50:9026)

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Article copyright: © Copyright 1974 American Mathematical Society

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